Experiments on routes to chaos in ball bearings

Abstract

The theoretical motion of a ball bearing has been studied in a previous paper. Using a control parameter, different routes to chaos were described. The aim of this paper is to study the experimental routes to chaos in a ball bearing and to confirm whether theoretical predictions of the phenomena are realistic.

An experimental test bench has been used and a numerical procedure has been proposed for observing Poincaré maps. As the control parameter varies the bearing clearly shows the appearance of instability in its motion. Two different routes to chaos are described as expected from the theory.

The first route is related to the first resonant frequency of the bearing. It is a sub-harmonic route. The second route, associated with the second resonant frequency, is a quasi-periodic route.

Introduction

Experiments in ball bearings are important for validating theories and bearing behaviour. To many users a ball bearing is considered as a black box containing balls, one ball retainer, some lubricant and sometimes seals. All parts are rotating except for one ring. Bearings are present in almost all rotating systems and are unknown to users unless something goes wrong. This is an interesting field for research to develop predictive methods to warn users before any damage in the case of bearing failure. An important literature deals with bearing responses in condition monitoring in machines. In Refs. [1], [2], the bearings vibrations are not only evaluated by Fourier spectrums and mean values but also with characteristic tools used in nonlinear dynamics such as phase portraits and Poincaré sections. An advanced work in this paper is to use these same tools to quantify the vibration levels and to analyse the internal dynamics of a ball bearing. The understanding of ball bearing behaviour implies a good knowledge of the load distribution of rolling element bearings, contact angles and the contact pressures between balls and raceways.

Some arrangements for analysing the internal behaviour of ball bearings are well known. The simplest method is to apply a thin copper layer onto the race-way, and then run the bearing for a short time under the desired load. After completely dismantling the bearing assembly the challenge is then to analyse the state of the remaining copper layer. The circumferential shape of the uncovered steel is an indication of the contact path. From this, the maximum loads and the unloaded area can then be located. In this experiment, the contact slip generates local wear of the materials. This wear depends on the traction coefficient that is a property of the lubricant but also the local temperature. In such experiments, one may obtain uncertain shapes and therefore the conclusions are not straightforward. A completely different method would be to modify parts of the bearing. A transparent and plastic outer ring is a useful way to obtain interferometry patterns which are characteristic of contact loads. This kind of analysis is specifically used to determine contact stresses. Finally, it is clear that there are still challenges in analysing the internal behaviour of bearings.

Fortunately, the safe dynamic motion of ball bearings is the most common behaviour encountered in industry. Fittings and loads are applied to the bearings and are generally appropriately defined in order to avoid high levels of vibrations, unexpected instabilities and low levels of reliability. In case the loads and fittings are not correctly set, the vibration level increases and the dynamic motion of the ball bearing becomes complicated.

Theoretical studies of bearing behaviour started in the early 1950s. At this time only quasi-static analysis was possible. Inertia and damping were neglected but the influence of clearance and loads was clearly stated. A kinematic study was proposed by Sunnersjo [3] who showed that vibration levels depended on inertia and speed. Some experiments were carried out to validate the proposed theory. The influence of the type of loads, radial clearance and rotational speed was then clearly demonstrated.

A nonlinear model of a ball bearing subject to a constant radial load was proposed by Fukata et al. [4]. It shows the influence of the rotating speed. Chaos-like, super-harmonic and sub-harmonic motions were analysed using time series and Poincaré maps. An increase of vibration levels around two specific rotation speeds was also noticeable.

In Ref. [5] a nonlinear model of a ball bearing with five degrees of freedom was proposed to study its stiffness under different kinds of loading. A three degree of freedom model was used in Ref. [6] and quasi-periodic and sub-harmonic routes were reported. In Ref. [7], a two degree of freedom model was used to simulate a radially loaded bearing. By modifying the damping factors of the model, different routes to chaos were noticeable and related to critical speeds of the ball bearing. Around the first critical speed the ball bearing shows instability and generates sub-harmonics of the ball pass frequency. Using the damping factor as a control parameter, it is possible to reach a chaotic region after an infinite number of bifurcations. Around the second critical speed, some combinations of the second resonant frequency and the ball pass frequency occur and generate a quasi-periodic motion. As the internal damping factor of the bearing decreases, the number of combinations of the two basic frequencies increases and rapidly overlaps, resulting in chaotic motion. Finally, the introduction of some over-sized balls in the bearing generates chaotic motions; this route was called the intermittent-like route.

A more recent study of the different kinds of motions in a nonlinear ball bearing model was proposed by Harsha in Ref. [8]. The bearing behaviour was studied with different levels of load and speed. The observations were similar to those in Ref. [7] with periodic, sub-harmonic and quasi-periodic Poincaré maps.

To describe these strange motions, tools specific to chaotic dynamics have to be introduced [9], [10], [11]. Fourier spectra are convenient for detecting sub- or super-harmonics of a component, also in the case of complete chaotic behaviour, but the quasi-periodic motion is impossible to detect except for the ideal case of two incommensurate components. Some recent studies, not associated with bearing motion, have used phase planes and Poincaré sections. The former just plots displacements as a function of themselves or their derivative. An extremely efficient technique is then to sample the phase plane points using a convenient clock frequency, in order to obtain a limited number of points. The resulting shape is an excellent tool to characterise sub-harmonic, quasi-periodic or chaotic motions.

Most studies introduced here are merely theoretical, and a lack of experiments is noticeable. An interesting paper, [12], deals with an experimental study of bearings in a rotor system. The work reported did not show chaotic behaviour but a period-doubling phenomenon is obvious.

The work of Ghafari et al. [1] confirmed the existence of chaos in healthy bearings, and the potential of chaos tools for the diagnosis of faults in rotating machinery. Another interesting paper is [2] in which faulty bearings are analysed using Poincaré maps. These are indications that the theoretical chaos tools provide an opportunity to demonstrate their power in industrial applications.

As a continuation of Ref. [7], an experiment was carried out in this paper to confirm our predictions of routes to chaos in ball bearings. The experiment is based on a test rig specially dedicated and an experimental procedure to generate projection of attractors. This paper presents the main results of this experiment.